Sketch the graphs y=4-x and `y=log_(10)x` on the same diagram.

Answer

To sketch the graphs of the equations \(y = 4 - x\) and \(y = \log_{10} x\) on the same diagram, we will follow these steps:

### Step 1: Find Points for the Linear Equation \(y = 4 - x\)

1. **Choose values for \(x\)**:
- Let \(x = 0\):
\[ y = 4 - 0 = 4 \quad \text{(Point: (0, 4))} \]
- Let \(x = 1\):
\[ y = 4 - 1 = 3 \quad \text{(Point: (1, 3))} \]
- Let \(x = 2\):
\[ y = 4 - 2 = 2 \quad \text{(Point: (2, 2))} \]
- Let \(x = 3\):
\[ y = 4 - 3 = 1 \quad \text{(Point: (3, 1))} \]
- Let \(x = 4\):
\[ y = 4 - 4 = 0 \quad \text{(Point: (4, 0))} \]

### Step 2: Find Points for the Logarithmic Equation \(y = \log_{10} x\)

1. **Choose values for \(x\)** (Note: \(x\) must be greater than 0 for the logarithm):
- Let \(x = 1\):
\[ y = \log_{10} 1 = 0 \quad \text{(Point: (1, 0))} \]
- Let \(x = 10\):
\[ y = \log_{10} 10 = 1 \quad \text{(Point: (10, 1))} \]
- Let \(x = 100\):
\[ y = \log_{10} 100 = 2 \quad \text{(Point: (100, 2))} \]

### Step 3: Plot the Points on a Graph

1. **Draw the axes**:
- Label the horizontal axis as \(x\) and the vertical axis as \(y\).

2. **Plot the points for \(y = 4 - x\)**:
- (0, 4), (1, 3), (2, 2), (3, 1), (4, 0)

3. **Plot the points for \(y = \log_{10} x\)**:
- (1, 0), (10, 1), (100, 2)

### Step 4: Draw the Lines and Curves

1. **For \(y = 4 - x\)**:
- Connect the points with a straight line. This line will have a negative slope and intercept the y-axis at 4.

2. **For \(y = \log_{10} x\)**:
- Draw a curve starting from the point (1, 0) and rising slowly as \(x\) increases. The curve approaches the y-axis but never touches it (asymptote at \(x = 0\)).

### Step 5: Identify the Intersection Point

1. **Find where the two graphs intersect**:
- Set \(4 - x = \log_{10} x\) and solve for \(x\) to find the intersection point.

### Final Graph

- You should now have a graph with a straight line for \(y = 4 - x\) and a logarithmic curve for \(y = \log_{10} x\). The intersection point can be approximated by checking values or using numerical methods.

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